A Comparison of Efficiency for Homogeneity of Variance Tests under Mixture Normal Distribution

Abstract

Background and Objectives: In statistical data analysis, the validity and reliability of conclusions derived from hypothesis testing depend on several crucial assumptions. One of the most important assumptions is the Homogeneity of Variance, which requires that the variance of the data in every studied population must be equal. This assumption is a necessary condition for widely used statistical tests, such as the t-test for comparing the means of two populations and F-test for comparing the means of three or more populations. Statisticians have developed several testing methods to verify this assumption. Traditional methods like Bartlett’s test are well-known for having high efficiency when the data follow a normal distribution but are extremely sensitive to violations of normality. Consequently, other tests with greater robustness have been proposed, such as Levene’s test (in both mean-based and median-based forms), the Non-parametric Levene’s test, O'Brien’s test, and the Fligner-Killeen test, each of which has different characteristics and efficiency depending on the nature of the data, such as skewness, kurtosis, and sample size.Currently, real-world data are often more complex than a single probability distribution, frequently appearing in the form of a mixture distribution. The mixture normal distribution is a very common probability distribution found in empirical research, such as in financial markets, medicine, and epidemiology. Using standard normal models is therefore insufficient for explaining these data. Studying the efficiency of homogeneity of variance tests under mixture distributions is critically necessary. This research, therefore, aims to study and compare the efficiency of six homogeneity of variance tests: Bartlett’s test, Mean-based Levene’s test, Median-based Levene’s test, Non-parametric Levene’s test, O'Brien’s test, and the Fligner-Killeen test, focusing on three population groups under a two-component mixture normal distribution, and considering the Type I error probability and statistical power criteria across various simulated situations.

Methodology: Data for the three groups were generated from a two-component mixture normal distribution with  mixing proportions (p ) were p=0.5 and p=0.8. The mean for all three groups was set to 8.8 with initial variances specified at 5, 10, and 20. Equal sample sizes were assigned to every group as (10,10,10), (20,20,20), (40,40,40), (60,60,60), (80,80,80), and (100,100,100). The variance ratios were set at 1:1:1, 1:1:2, 1:1:4 and 1:2:4. To consider the ability to control Type I error, Bradley’s liberal criterion was used at a significance level of 0.05. A test considered capable of controlling type I error must have an estimated type I error probability within the range of [0.025, 0.075]. In considering statistical power, only tests that passed Bradley’s criterion were considered; an efficient test must have an estimated statistical power of not less than 0.80, and the method with the highest value is considered the most efficient.

Main Results: The study demonstrates that the efficiency of all six testing methods differs depending on the mixing proportion and sample size. When considering the ability to control type I error: in the case where p=0.8  , the Median-based Levene’s test, Non-parametric Levene’s test, O’Brien’s test, and Fligner-Killeen test were robust, being able to control Type I error across all sample sizes, while Bartlett’s test required a sample size of 60 or more to maintain control. The Mean-based Levene’s test could not control the error in any situation. In the case where p=0.5 , the Median-based Levene’s test, Non-parametric Levene’s test, and O’Brien’s test could control Type I error across all sample sizes. The Mean-based Levene’s test began to maintain control when . However, Bartlett’s test and the Fligner-Killeen test could not control the error under this mixing proportion. In considering the estimated statistical power, only tests that could control Type I error were examined. It was found that the estimated statistical power increases according to sample size and the difference in variances. At p=0.8 : All four methods capable of controlling Type I error required a sample size of   for the 1:1:2 ratio and  for the 1:1:4 ratio to achieve statistical power higher than 0.8. Bartlett’s test provided the highest statistical power in conditions where it could control Type I error. At p=0.5, O’Brien’s test was the most prominent, providing statistical power higher than 0.8 when  at a 1:1:2 ratio and   at a 1:1:4 ratio, which outperformed all forms of Levene’s test.

Conclusions : The results of this study: 1. Bartlett’s Test: Although highly efficient in some conditions ( p=0.8 and large sample size), it cannot control Type I error when the data have a mixture distribution with equal mixing proportions, making it a high risk for application with complex data. 2. Mean-based Levene’s Test: Highly sensitive to mixing proportions and only applicable when the mixture distribution is balanced and the sample size is moderate or larger. 3. Fligner-Killeen Test: Performs well only in cases where the mixing proportion p=0.8 . 4. O’Brien’s Test: This method is identified as the most suitable and efficient overall. It not only successfully controlled the Type I error under all studied conditions but also provided the highest statistical power in nearly every scenario, particularly with small to moderate sample sizes.